A store starts a week with at least 3 PCs. The demand per week is estimated at 0 with probability .15, 1 with probability .2, 2 with probability .35, 3 with probability .25, and 4 with probability .05. Unfilled demand is backlogged. The store’s policy is to place an order for delivery at the start of the following week whenever the inventory level drops below 3 PCs. The new replenishment always brings the stock back to 5 PCs.

(a) Express the situation as a Markov chain.

(b) Suppose that the week starts with 4 PCs. Determine the probability that an order will be placed at the end of two weeks.

(c) Determine the long-run probability that no order will be placed in any week.

(d) If the fixed cost of placing an order is $200, the holding cost per PC per week is

$5, and the penalty cost per shortage PC per week is $20, determine the expected inventory cost per week.